Leray-type
Regularizations of the Burgers and the
Isentropic Euler Equations

We start from the Burgers equation
vt + vvx = 0
and investigate a smoothing mechanism that
replaces the convective velocity v in the
nonlinear term by a smoother velocity field u.
This type of regularization was first proposed
in 1934 by Leray, who applied it in the context
of the incompressible Navier-Stokes equations.
We show strong analytical and numerical
indication that the Leray smoothing procedure
yields a valid regularization of the Burgers
equation.

We also study the stability of the front
traveling waves. The front stability results
show that the regularized equation mirrors the
physics of rarefaction and shock waves in the
Burgers equation.

Finally, we apply the Leray regularization to
the isentropic Euler equations and use the
weakly nonlinear geometrical optics (WNGO)
asymptotic theory to analyze the resulting
system. As it turns out, the Leray
procedure regularizes the Euler equations only
in special cases. We further investigate these
cases using Riemann invariants techniques.

Algebras of
Time-Frequency Shift Operators with Applications
to Wireless Communications

In this talk I present results on various Banach
*-algebras of time-frequency shift operators
with absolutely summable coefficients, and two
applications. The L1 theory contains
noncommutative versions of the Wiener lemma with
various norm estimates. The L2 theory
is built around an explicit formula of the
faithful trace for this algebra. The Zak
transform turns out to be an effective
computational tool for practical cases.

One
application is the Heil-Ramanathan-Topiwala
conjecture that states that finitely many
time-frequency shifts of one L2
function are linearly independent. This turns to
be equivalent to the absence of eigenspectrum
for finite linear combinations of time-frequency
shifts. I will prove a special case of this
conjecture.

Next I present two applications in Wireless
communications. One application is the channel
equalization problem. I will show how to use the
Wiener lemma and the Zak transform to
effectively compute this inverse. A second
application concerns canonical communication
channels. The original canonical channel model
allows for designing the Rake receiver. Recently
Sayeed and Aazhang obtained a time-frequency
canonical model and designed its Rake receiver.
I will comment on its operator algebra meaning
and ways to generalize it to other canonical
models using time-scale or frequency-scale
shifts.

We present a novel inversion algorithm for
electrical impedance tomography in two
dimensions, based on a model reduction approach.

The reduced models are resistor networks that
arise in five point stencil discretizations of
the elliptic partial differential equation
satisfied by the electric potential, on adaptive
grids that are computed as part of the problem.
We prove the unique solvability of the model
reduction problem for a broad class of
measurements of the Dirichlet to Neumann map.
The size of the networks (reduced models) is
limited by the precision of the measurements.
The resulting grids are naturally refined near
the boundary, where we make the measurements and
where we expect better resolution of the images.
To determine the unknown conductivity, we use
the resistor networks to define a nonlinear
mapping of the data, that behaves as an
approximate inverse of the forward map.

Then, we propose an efficient Newton-type
iteration for finding the conductivity, using
this map. We also show how to incorporate
apriori information about the conductivity in
the inversion scheme.

We consider decay of solutions of the Cauchy
Problem for various fields in the Kerr (rotating
Black Hole) geometry. We discuss the formulation
of the problem in terms of the Teukolsky
equation, a single second-order PDE depending on
a real parameter s, the "spin". For various
values of s, the Teukolsky equation describes
different fields in the Kerr geometry:
s=0,1/2,1, 2, correspond,respectively, to scalar
waves, Dirac's equation, Maxwell's equations,
and gravitational waves. We discuss our results
for s=0,1/2, as well as our rigorous proof of
Penrose's proposal (1969) for energy extraction
from a Kerr BH. In the case of a Schwarzschild
(non-rotating) BH, we discuss our decay results,
and hence stability of this BH, for all spin.

The Ck- extension problem posed by
H. Whitney in 1932 is to provide conditions for
a function f defined on a closed set A in Rn,
to be a restriction of a Ck-smooth
function F on the entire space Rn.
For n=1 the answer is (roughly) given by the
existing of a uniform bound for all the finite
differences up to order k of f on A. In higher
dimensions this problem was solved only very
recently by Ch. Fefferman. There is a natural
generalization of the Whitney's problem to the
high-order data (Whitney fields). We show that
the Ck-extension from regular
lattices is closely related to the Hermite
polynomial approximation of high-order data. The
last problem can be studied with the tools of
real Semi-Algebraic Geometry.

Finally, we discuss some high order numerical
schemes, based on the Whitney extension of the
high-order lattice data.

Abstract: When a plane shock hits a wedge head
on, it experiences a reflection-diffraction
process and then a self-similar reflected shock
moves outward as the original shock moves
forward in time. The complexity of reflection
picture was first reported by Ernst Mach in
1878, and experimental, computational, and
asymptotic analysis has shown that various
patterns of reflected shocks may occur,
including regular and Mach reflection. However,
most fundamental issues for shock reflection
have not been understood, including the
transition of the different patterns of shock
reflection, and there had been no rigorous
mathematical result on the global existence and
structural stability of shock reflection,
especially for potential flow which has widely
been used in aerodynamics.

In this talk we will start with various shock
reflection phenomena and their fundamental
scientific issues. Then we will describe how the
shock reflection problems can be formulated into
free boundary problems for nonlinear partial
differential equations of mixed-composite type.
Finally we will discuss some recent developments
in attacking the shock reflection problems,
including our results with M. Feldman on the
global existence and stability of solutions of
shock reflection by large-angle wedges for
potential flow. The approach includes techniques
to handle free boundary problems, degenerate
elliptic equations, and corner singularities
when free boundaries meet degenerate elliptic
curves.

Recent theoretical and numerical results on
intermittency in hydrodynamic turbulence are
described, with special emphasis on the
Lagrangian evolution.

First, we derive the advected delta-vee system.
This simple dynamical system deals with the
Lagrangian evolution of two-point velocity and
scalar increments in turbulence and shows how
many known trends in turbulence can be simply
understood from the proposed projection of the
self-stretching effect coming from the nonlinear
advective term. More detailed statistical
information can be obtained from a model for the
full velocity gradient tensor that uses a
closure for the pressure Hessian and viscous
terms.

Finally, we will describe efforts to use these
insights in the generation of synthetic,
multi-scale 3D vector fields with non-Gaussian
statistics that reproduce many of know behavior
of turbulence.

Professor Dionisios Margetis, Department
of Mathematics, University of Maryland

From microscopic physics
to continuum laws for crystal surfaces: Progress
and challenges

In materials modeling the starting point
("truth") is atomistic, or takes the form of
discrete schemes, by which evolution laws must
be determined for the macroscale. In this talk I
will focus on the evolution of crystal surfaces
as a prototypical case of modeling across the
scales, with implications in the design of novel
solid-state devices.
I will discuss recent progress in three
directions:

(i) Microscopic details of crystals enter free
boundary problems, and affect evolution at the
macroscale.
(ii) Continuum laws give signatures of
instabilities that are due to the discrete
nature of crystal surfaces.
(iii) Material parameters in a continuum
description are determined as appropriate limits
of microscopic, kinetic processes.

I will also discuss challenges and open
questions that arise in this context.

Gravitational wave detectors like LIGO are
poised to begin detecting signals. One of the
prime scientific goals is to detect waves from
the coalescence and merger of black holes in
binary systems. Confronting such signals with
the predictions of Einstein's General Theory of
Relativity will be the first real strong-field
test of the theory. I will explain why
multidomain spectral methods are ideally suited
to tackle this problem.
Until very recently, theorists were unable to
calculate what the theory actually predicts. I
will describe recent breakthroughs that have
occurred and that have set things up for an epic
confrontation of theory and experiment.

Microbes live in
fluctuating environments that are often limiting
for growth. They have evolved several
sophisticated mechanisms to sense changes in
important environmental parameters such as light
and nutrients. Most bacteria also have complex
appendages that allow them to move, so they can
swim or crawl into optimal conditions. This
combination of sensing changes in the immediate
environment and transducing these changes to the
motion organisms, allows for movement in a
particular direction: a phenomenon known as
''chemotaxis'' or ''phototaxis''.

Using time-lapse video microscopy we have
monitored the movement of Cyanobacteria (which
are phototaxis, i.e., bacteria that move towards
light). These movies suggest that single cells
are able to move directionally but at the same
time, the group dynamics is equally important.
The various patterns of movement that we observe
appear to be a complex function of cell density,
surface properties and genotype. Very little is
known about the interactions between these
parameters.

In this talk, we will present a hierarchy of new
models for describing the motion of phototaxis
that were constructed based on the experimental
observations. The first model is a stochastic
model that describes the locations of bacteria,
the group dynamics, and the interaction between
the bacteria and the medium in which it resides.
The second model is a new multi-particle system
that is obtained from a discretization of the
first model. Our third model is obtained as the
continuum limit of
the second model, and as such it is a system of
nonlinear PDEs. Our main theorems clarify the
sense in which the system of PDEs can be
considered as the limit dynamics of the
multi-particle system. We conclude with several
numerical simulations that demonstrate the
properties of our models.

This is a joint work with Devaki Bhaya
(Department of Plant Biology, Carnegie
Institute) and Tiago Requeijo (Math, Stanford)

We present recent numerical methods and
simulations of time dependent NLS with
nonlinearities of the local type ("cubic NLS")
or/and of the nonlocal type ("Hartree/Poisson
equation") and of Davey Stewartson equations.
Particular interest is laid on simulations of
"blow up" of the critical (= 2-d cubic) NLS (*),
depending on parameters of the data / equation,
where a conjecture on monotonicity of the blow
up time (Fibich) is shown not to hold. Also we
show the Schroedinger-Poisson-X-alpha model,
where we investigate properties of
semi-classical asymptotics. We briefly present
the equations, analysis and the numerical
methods (time split spectral scheme and
relaxation scheme implemented on a fixed grid on
a parallel machine, thus allowing for 3-d
simulations with up to 1000 points in each
direction) and instructive simulation results.

In this talk I present the implementation of
adaptive central schemes on unstructured
tetrahedral grids for approximating nonlinear
hyperbolic conservation laws in 3D. The mesh
adaptation algorithm is coupled to a cell
vertex, finite volume scheme of the MUSCL type,
employing a second order central schemes. First
and second order central central schemes on
staggered grids see figs. 1,2 were introduced by
Nessyahu and Tadmor [7] in one spatial dimension
and generalized to unstructured grids in two
space dimensions by Arminjon, Viallon and
Madrane [1], and in three dimensions by
Arminjon, Madrane and StCyr [2,3,4]. In
addition, a posteriori error estimates were
obtained, by interpreting the central scheme on
staggered grids as a upwind finite volume scheme
in conservation form [5]. In this contribution
we extend the idea of [5,6] to 3D to derive
appropriate refinement indicators.
The adaptive scheme is then used to compute the
shock waves around an NACA0012 airfoil for
subsonic post-shock flow.

A Conjectured Hierarchy
of Length Scales in a Generalization of the
Navier-Stokes-alpha Equation for Turbulent Fluid
Flow

We present a continuum-mechanical formulation
and generalization of the Navier–Stokes-α theory
based on a general framework for fluid-dynamical
theories with gradient dependencies. Our flow
equation involves two additional
problem-dependent length scales α and β. The
first of these scales enters the theory through
the internal kinetic energy, per unit mass,
α2|D|2, where D is the symmetric part of the
gradient of the filtered velocity. The remaining
scale is associated with a dissipative
hyperstress which depends linearly on the
gradient of the filtered vorticity. When α and β
are equal, our flow equation reduces to the
Navier–Stokes-α equation. In contrast to the
original derivation of the Navier–Stokes-α
equation, which relies on Lagrangian averaging,
our formulation delivers boundary conditions.
For a confined flow, our boundary conditions
involve an additional length scale
l characteristic of the eddies found near
walls. Based on a comparison with direct
numerical simulations for fully-developed
turbulent flow in a rectangular channel of
height 2h, we find that α/β ~ Re0.470
and l/h ~ Re−0.772 , where Re
is the Reynolds number. The first result, which
arises as a consequence of identifying the
internal kinetic energy with the turbulent
kinetic energy, indicates that the choice α = β
required to reduce our flow equation to the
Navier–Stokes-α equation is likely to be
problematic. The second result evinces the
classical scaling relation η/L ~ Re−3/4
for the ratio of the Kolmogorov microscale η to
the integral length scale L. The numerical data
also suggests that l≤
β.
We are therefore led to conjecture a tentative
hierarchy, l
≤ β < α, involving the three length
scales entering our theory.

The Surprising
Structure of Gaussian Point
Clouds and its Implications for Signal
Processing

We will explore connections between
the structure of high-dimensional convex
polytopes and information acquisition for
compressible signals. A classical result in the
field of convex polytopes is that if N points
are distributed Gaussian iid at random in
dimension n<<N, then only order (log N)^n of the
points are vertices of their convex hull. Recent
results show that
provided n grows slowly with N, then with high
probability all of the points are vertices of
its convex hull. More surprisingly, a rich
"neighborliness" structure emerges in the faces
of the convex hull. One implication of this
phenomenon is that an N-vector with k non-zeros
can be recovered computationally efficiently
from only n random projections with n=2e k log(N/n).
Alternatively, the best k-term approximation of
a signal in any basis can be recovered from 2e k
log(N/n) non-adaptive measurements, which is
within a log factor of the optimal rate
achievable for adaptive sampling. Additional
implications for randomized error correcting
codes will be presented.